Emergence of Elastostatic Strain-Gradient Effects from Topological Optimization ; Émergence d'effets de strain-gradient en élastostatique par optimisation topologique

International audience ; Practical examples of microstructures producing so-called strain-gradient effects in elastostatics are not very common. Therefore, the topological optimization of strain-gradient effects for two-dimensional periodic media in elastostaic regime is considered in the present article, in order to obtain new microstructures featuring these effects. The periodic unit cell is made of a mixture of stiff and soft materials. The optimization process is carried out through the solution of a shape optimization problem. On the one hand, the design variable is the distribution of material inside the unit cell. On the other hand, the shape functional depends on this distribution through related first and second-order homogenized tensors, which are defined from a homogenization scheme based on the two-scale asymptotic expansion. The adopted method used to tackle numerically the optimization problem is based on the topological derivatives of the homogenized tensors, which measure how the homogenized elasticity tensors change when a small circular inclusion endowed with different material property from the background is introduced at the microscopic level. With this approach, new microstructures with macroscopic strain-gradient effects are obtained. In particular, we retrieve well-known microstructures such as the pantographic material.